A particle is projected up an inclined plane of inclination $\mathrm{\beta }$ at an elevation $\mathrm{\alpha }$ to the horizontal. Find the ratio between $\tan \mathrm{\alpha }\text{ and }\tan \mathrm{\beta }$, if the particle strikes the plane horizontally.

$\mathrm{T}=\frac{2\mathrm{usin}(\mathrm{\alpha }-\mathrm{\beta })}{\mathrm{gcos}\mathrm{\beta }}$             …(i)

Now we shall consider the motion of particle along OA. 

When the particle strikes the plane horizontally 

In this case, $\mathrm{t}=\frac{2\mathrm{usin}(\mathrm{\alpha }-\mathrm{\beta })}{\mathrm{gcos}\mathrm{\beta }}$

and $0=\mathrm{usin}\mathrm{\alpha }-\mathrm{gt}$

$\mathrm{usin}\mathrm{\alpha }=\mathrm{g}\left[\frac{2\mathrm{usin}(\mathrm{\alpha }-\mathrm{\beta })}{\mathrm{gcos}\mathrm{\beta }}\right]$

$2\tan \mathrm{\beta }=\tan \mathrm{\alpha }\Rightarrow \tan \mathrm{\alpha }:\tan \mathrm{\beta }=2$